Physics 1 - Class XII

8/18/2019 Physics 1 - Class XII

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CONTENTS

F OREWORD v

PREFACE xi

CHAPTER ONE

E LECTRIC CHARGES AND F IELDS

1.1 Introduction 1

1.2 Electric Charges 1

1.3 Conductors and Insulators 5

1.4 Charging by Induction 6

1.5 Basic Properties of Electric Charge 8

1.6 Coulomb’s Law 10

1.7 Forces between Multiple Charges 15

1.8 Electric Field 18

1.9 Electric Field Lines 23

1.10 Electric Flux 25

1.11 Electric Dipole 27

1.12 Dipole in a Uniform External Field 31

1.13 Continuous Charge Distribution 32

1.14 Gauss’s Law 331.15 Application of Gauss’s Law 37

CHAPTER TWO

E LECTROSTATIC POTENTIAL AND C APACITANCE

2.1 Introduction 51

2.2 Electrostatic Potential 53

2.3 Potential due to a Point Charge 54

2.4 Potential due to an Electric Dipole 55

2.5 Potential due to a System of Charges 57

2.6 Equipotential Surfaces 602.7 Potential Energy of a System of Charges 61

2.8 Potential Energy in an External Field 64

2.9 Electrostatics of Conductors 67

2.10 Dielectrics and Polarisation 71

2.11 Capacitors and Capacitance 73

2.12 The Parallel Plate Capacitor 74

2.13 Effect of Dielectric on Capacitance 75

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2.14 Combination of Capacitors 78

2.15 Energy Stored in a Capacitor 80

2.16 Van de Graaff Generator 83

CHAPTER THREE

CURRENT E LECTRICITY

3.1 Introduction 93

3.2 Electric Current 93

3.3 Electric Currents in Conductors 94

3.4 Ohm’s law 95

3.5 Drift of Electrons and the Origin of Resistivity 97

3.6 Limitations of Ohm’s Law 101

3.7 Resistivity of various Materials 101

3.8 Temperature Dependence of Resistivity 1033.9 Electrical Energy, Power 105

3.10 Combination of Resistors — Series and Parallel 107

3.11 Cells, emf, Internal Resistance 110

3.12 Cells in Series and in Parallel 113

3.13 Kirchhoff’s Laws 115

3.14 Wheatstone Bridge 118

3.15 Meter Bridge 120

3.16 Potentiometer 122

CHAPTER FOUR

MOVING CHARGES AND M AGNETISM

4.1 Introduction 132

4.2 Magnetic Force 133

4.3 Motion in a Magnetic Field 137

4.4 Motion in Combined Electric and Magnetic Fields 140

4.5 Magnetic Field due to a Current Element, Biot-Savart Law 143

4.6 Magnetic Field on the Axis of a Circular Current Loop 145

4.7 Ampere’s Circuital Law 147

4.8 The Solenoid and the Toroid 150

4.9 Force between Two Parallel Currents, the Ampere 154

4.10 Torque on Current Loop, Magnetic Dipole 1574.11 The Moving Coil Galvanometer 163

CHAPTER FIVE

M AGNETISM AND M ATTER


5.2 The Bar Magnet 174

xiv

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5.3 Magnetism and Gauss’s Law 181

5.4 The Earth’s Magnetism 185

5.5 Magnetisation and Magnetic Intensity 189

5.6 Magnetic Properties of Materials 191

5.7 Permanent Magnets and Electromagnets 195

CHAPTER SIX

E LECTROMAGNETIC INDUCTION


6.2 The Experiments of Faraday and Henry 205

6.3 Magnetic Flux 206

6.4 Faraday’s Law of Induction 207

6.5 Lenz’s Law and Conservation of Energy 210

6.6Motional Electromotive Force 212

6.7 Energy Consideration: A Quantitative Study 215

6.8 Eddy Currents 218

6.9 Inductance 219

6.10 AC Generator 224

CHAPTER SEVEN

A LTERNATING CURRENT


7.2 AC Voltage Applied to a Resistor 234

7.3 Representation of AC Current and Voltage by Rotating Vectors — Phasors 237

7.4 AC Voltage Applied to an Inductor 237

7.5 AC Voltage Applied to a Capacitor 241

7.6 AC Voltage Applied to a Series LCR Circuit 244

7.7 Power in AC Circuit: The Power Factor 252

7.8 LC Oscillations 255

7.9 Transformers 259

CHAPTER EIGHT

E LECTROMAGNETIC W AVES


8.2 Displacement Current 2708.3 Electromagnetic Waves 274

8.4 Electromagnetic Spectrum 280

ANSWERS 288

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1.1 INTRODUCTION All of us have the experience of seeing a spark or hearing a crackle when we take off our synthetic clothes or sweater, particularly in dry weather. This is almost inevitable with ladies garments like a polyester saree. Have you ever tried to find any explanation for this phenomenon? Another common example of electric discharge is the lightning that we see in thesky during thunderstorms. We also experience a sensation of an electricshock either while opening the door of a car or holding the iron bar of a

bus after sliding from our seat. The reason for these experiences isdischarge of electric charges through our body, which were accumulateddue to rubbing of insulating surfaces. You might have also heard that this is due to generation of static electricity. This is precisely the topic weare going to discuss in this and the next chapter. Static means anythingthat does not move or change with time. Electrostatics deals with the study of forces, fields and potentials arising from static charges .

1.2 ELECTRIC CHARGEHistorically the credit of discovery of the fact that amber rubbed with

wool or silk cloth attracts light objects goes to Thales of Miletus, Greece,

around 600 BC. The name electricity is coined from the Greek wordelektron meaning amber . Many such pairs of materials were known which

Chapter One

ELECTRIC CHARGES

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on rubbing could attract light objects

like straw, pith balls and bits of papers. You can perform the following activity at home to experience such an effect.Cut out long thin strips of white paper and lightly iron them. Take them near a

TV screen or computer monitor. You willsee that the strips get attracted to thescreen. In fact they remain stuck to thescreen for a while.

It was observed that if two glass rodsrubbed with wool or silk cloth are

brought close to each other, they repeleach other [Fig. 1.1(a)]. The two strandsof wool or two pieces of silk cloth, with

which the rods were rubbed, also repeleach other. However, the glass rod and

wool attracted each other. Similarly, two plastic rods rubbed with cat’sfur repelled each other [Fig. 1.1(b)] but attracted the fur. On the other hand, the plastic rod attracts the glass rod [Fig. 1.1(c)] and repel the silk or wool with which the glass rod is rubbed. The glass rod repels the fur.

If a plastic rod rubbed with fur is made to touch two small pith balls(now-a-days we can use polystyrene balls) suspended by silk or nylonthread, then the balls repel each other [Fig. 1.1(d)] and are also repelled

by the rod. A similar effect is found if the pith balls are touched with a glass rod rubbed with silk [Fig. 1.1(e)]. A dramatic observation is that a pith ball touched with glass rod attracts another pith ball touched withplastic rod [Fig. 1.1(f )].

These seemingly simple facts were established from years of effortsand careful experiments and their analyses. It was concluded, after many careful studies by different scientists, that there were only two kinds of an entity which is called the electric charge . We say that the bodies likeglass or plastic rods, silk, fur and pith balls are electrified. They acquirean electric charge on rubbing. The experiments on pith balls suggestedthat there are two kinds of electrification and we find that (i) like charges

repel and (ii) unlike charges attract each other. The experiments alsodemonstrated that the charges are transferred from the rods to the pith balls on contact. It is said that the pith balls are electrified or are charged by contact. The property which differentiates the two kinds of charges iscalled the polarity of charge.

When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge. This is true for any pair of objects that are rubbed to be electrified. Now if the electrifiedglass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other. They also do not attract or repel other light objects as they did on being electrified.

Thus, the charges acquired after rubbing are lost when the charged

bodies are brought in contact. What can you conclude from theseobservations? It just tells us that unlike charges acquired by the objects

FIGURE 1.1 Rods and pith balls: like charges repel andunlike charges attract each other.

I n t e r a c t i v e

a n i m a t i o n

o n

s i m p l e

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e x

p e r i m e n t s :

h t t p : / / e p h y s i c s . p h y s i c s . u c l a . e d u / t r a v o l t a g e / H T M L /

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neutralise or nullify each other’s effect. Therefore the charges were named

as positive and negative by the American scientist Benjamin Franklin. We know that when we add a positive number to a negative number of the same magnitude, the sum is zero. This might have been thephilosophy in naming the charges as positive and negative. By convention,the charge on glass rod or cat’s fur is called positive and that on plasticrod or silk is termed negative. If an object possesses an electric charge, it is said to be electrified or charged. When it has no charge it is said to beneutral.

UNIFICATION OF ELECTRICITY AND MAGNETISM

In olden days, electricity and magnetism were treated as separate subjects. Electricity dealt with charges on glass rods, cat’s fur, batteries, lightning, etc., while magnetismdescribed interactions of magnets, iron filings, compass needles, etc. In 1820 Danishscientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle. Ampere and Faraday supported this observation

by saying that electric charges in motion produce magnetic fields and moving magnetsgenerate electricity. The unification was achieved when the Scottish physicist Maxwelland the Dutch physicist Lorentz put forward a theory where they showed theinterdependence of these two subjects. This field is called electromagnetism . Most of thephenomena occurring around us can be described under electromagnetism. Virtually every force that we can think of like friction, chemical force between atoms holding thematter together, and even the forces describing processes occurring in cells of living

organisms, have its origin in electromagnetic force. Electromagnetic force is one of thefundamental forces of nature.

Maxwell put forth four equations that play the same role in classical electromagnetismas Newton’s equations of motion and gravitation law play in mechanics. He also arguedthat light is electromagnetic in nature and its speed can be found by making purely electric and magnetic measurements. He claimed that the science of optics is intimately related to that of electricity and magnetism.

The science of electricity and magnetism is the foundation for the modern technologicalcivilisation. Electric power, telecommunication, radio and television, and a wide variety of the practical appliances used in daily life are based on the principles of this science.

Although charged particles in motion exert both electric and magnetic forces, in theframe of reference where all the charges are at rest, the forces are purely electrical. Youknow that gravitational force is a long-range force. Its effect is felt even when the distance

between the interacting particles is very large because the force decreases inversely asthe square of the distance between the interacting bodies. We will learn in this chapter that electric force is also as pervasive and is in fact stronger than the gravitational force

by several orders of magnitude (refer to Chapter 1 of Class XI Physics Textbook).

A simple apparatus to detect charge on a body is the gold-leaf electroscope [Fig. 1.2(a)]. It consists of a vertical metal rod housed in a

box, with two thin gold leaves attached to its bottom end. When a chargedobject touches the metal knob at the top of the rod, charge flows on to

the leaves and they diverge. The degree of divergance is an indicator of the amount of charge.

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Students can make a simple electroscope as

follows [Fig. 1.2(b)]: Take a thin aluminium curtainrod with ball ends fitted for hanging the curtain. Cut out a piece of length about 20 cm with the ball at one end and flatten the cut end. Take a large bottlethat can hold this rod and a cork which will fit in theopening of the bottle. Make a hole in the cork sufficient to hold the curtain rod snugly. Slide therod through the hole in the cork with the cut end onthe lower side and ball end projecting above the cork.Fold a small, thin aluminium foil (about 6 cm inlength) in the middle and attach it to the flattened

end of the rod by cellulose tape. This forms the leavesof your electroscope. Fit the cork in the bottle withabout 5 cm of the ball end projecting above the cork.

A paper scale may be put inside the bottle in advanceto measure the separation of leaves. The separationis a rough measure of the amount of charge on theelectroscope.

To understand how the electroscope works, usethe white paper strips we used for seeing theattraction of charged bodies. Fold the strips into half so that you make a mark of fold. Open the strip andiron it lightly with the mountain fold up, as shown

in Fig. 1.3. Hold the strip by pinching it at the fold. You would notice that the two halves move apart.

This shows that the strip has acquired charge on ironing. When you foldit into half, both the halves have the same charge. Hence they repel eachother. The same effect is seen in the leaf electroscope. On charging thecurtain rod by touching the ball end with an electrified body, charge istransferred to the curtain rod and the attached aluminium foil. Both thehalves of the foil get similar charge and therefore repel each other. Thedivergence in the leaves depends on the amount of charge on them. Let us first try to understand why material bodies acquire charge.

You know that all matter is made up of atoms and/or molecules.

Although normally the materials are electrically neutral, they do containcharges; but their charges are exactly balanced. Forces that hold themolecules together, forces that hold atoms together in a solid, the adhesiveforce of glue, forces associated with surface tension, all are basically electrical in nature, arising from the forces between charged particles.

Thus the electric force is all pervasive and it encompasses almost eachand every field associated with our life. It is therefore essential that welearn more about such a force.

To electrify a neutral body, we need to add or remove one kind of charge. When we say that a body is charged, we always refer to thisexcess charge or deficit of charge. In solids, some of the electrons, beingless tightly bound in the atom, are the charges which are transferred

from one body to the other. A body can thus be charged positively by losing some of its electrons. Similarly, a body can be charged negatively

FIGURE 1.2 Electroscopes: (a) The gold leaf electroscope, (b) Schematics of a simple

electroscope.

FIGURE 1.3 Paper stripexperiment.

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electric current from the power station and the third is earthed by

connecting it to the buried metal plate. Metallic bodies of the electricappliances such as electric iron, refrigerator, TV are connected to theearth wire. When any fault occurs or live wire touches the metallic body,the charge flows to the earth without damaging the appliance and without causing any injury to the humans; this would have otherwise beenunavoidable since the human body is a conductor of electricity.

1.4 CHARGING BY INDUCTION When we touch a pith ball with an electrified plastic rod, some of thenegative charges on the rod are transferred to the pith ball and it alsogets charged. Thus the pith ball is charged by contact . It is then repelled

by the plastic rod but is attracted by a glass rod which is oppositely charged. However, why a electrified rod attracts light objects, is a question

we have still left unanswered. Let us try to understand what could behappening by performing the following experiment.(i) Bring two metal spheres, A and B, supported on insulating stands,

in contact as shown in Fig. 1.4(a).(ii) Bring a positively charged rod near one of the spheres, say A, taking

care that it does not touch the sphere. The free electrons in the spheresare attracted towards the rod. This leaves an excess of positive chargeon the rear surface of sphere B. Both kinds of charges are bound inthe metal spheres and cannot escape. They, therefore, reside on thesurfaces, as shown in Fig. 1.4(b). The left surface of sphere A, has an

excess of negative charge and the right surface of sphere B, has anexcess of positive charge. However, not all of the electrons in the sphereshave accumulated on the left surface of A. As the negative chargestarts building up at the left surface of A, other electrons are repelled

by these. In a short time, equilibrium is reached under the action of force of attraction of the rod and the force of repulsion due to theaccumulated charges. Fig. 1.4(b) shows the equilibrium situation.

The process is called induction of charge and happens almost instantly. The accumulated charges remain on the surface, as shown,till the glass rod is held near the sphere. If the rod is removed, thecharges are not acted by any outside force and they redistribute totheir original neutral state.

(iii) Separate the spheres by a small distance while the glass rod is stillheld near sphere A, as shown in Fig. 1.4(c). The two spheres are foundto be oppositely charged and attract each other.

(iv) Remove the rod. The charges on spheres rearrange themselves asshown in Fig. 1.4(d). Now, separate the spheres quite apart. Thecharges on them get uniformly distributed over them, as shown inFig. 1.4(e).In this process, the metal spheres will each be equal and oppositely

charged. This is charging by induction . The positively charged glass roddoes not lose any of its charge, contrary to the process of charging by contact.

When electrified rods are brought near light objects, a similar effect

takes place. The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object.

FIGURE 1.4 Charging by induction.

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[This happens even when the light object is not a conductor. The

mechanism for how this happens is explained later in Sections 1.10 and2.10.] The centres of the two types of charges are slightly separated. Weknow that opposite charges attract while similar charges repel. However,the magnitude of force depends on the distance between the chargesand in this case the force of attraction overweighs the force of repulsion.

As a result the particles like bits of paper or pith balls, being light, arepulled towards the rods.

Example 1.1 How can you charge a metal sphere positively without touching it?

Solution Figure 1.5(a) shows an uncharged metallic sphere on aninsulating metal stand. Bring a negatively charged rod close to themetallic sphere, as shown in Fig. 1.5(b). As the rod is brought closeto the sphere, the free electrons in the sphere move away due torepulsion and start piling up at the farther end. The near end becomespositively charged due to deficit of electrons. This process of chargedistribution stops when the net force on the free electrons inside themetal is zero. Connect the sphere to the ground by a conducting wire. The electrons will flow to the ground while the positive chargesat the near end will remain held there due to the attractive force of the negative charges on the rod, as shown in Fig. 1.5(c). Disconnect the sphere from the ground. The positive charge continues to beheld at the near end [Fig. 1.5(d)]. Remove the electrified rod. Thepositive charge will spread uniformly over the sphere as shown in

Fig. 1.5(e).

FIGURE 1.5

In this experiment, the metal sphere gets charged by the processof induction and the rod does not lose any of its charge.

Similar steps are involved in charging a metal sphere negatively by induction, by bringing a positively charged rod near it. In thiscase the electrons will flow from the ground to the sphere when thesphere is connected to the ground with a wire. Can you explain why?

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1.5 B ASIC PROPERTIES OF ELECTRIC CHARGE We have seen that there are two types of charges, namely positive andnegative and their effects tend to cancel each other. Here, we shall now describe some other properties of the electric charge.

If the sizes of charged bodies are very small as compared to thedistances between them, we treat them as point charges . All thecharge content of the body is assumed to be concentrated at one point in space.

1.5.1 Additivity of charges

We have not as yet given a quantitative definition of a charge; we shall

follow it up in the next section. We shall tentatively assume that this can be done and proceed. If a system contains two point charges q 1 and q 2,the total charge of the system is obtained simply by adding algebraically q 1 and q 2 , i.e., charges add up like real numbers or they are scalars likethe mass of a body. If a system contains n charges q 1, q 2, q 3, …, q n, thenthe total charge of the system is q 1 + q 2 + q 3 + … + q n . Charge hasmagnitude but no direction, similar to the mass. However, there is onedifference between mass and charge. Mass of a body is always positive

whereas a charge can be either positive or negative. Proper signs have to be used while adding the charges in a system. For example, thetotal charge of a system containing five charges +1, +2, –3, +4 and –5,in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in thesame unit.

1.5.2 Charge is conserved

We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed. A picture of particles of electriccharge enables us to understand the idea of conservation of charge. When

we rub two bodies, what one body gains in charge the other body loses. Within an isolated system consisting of many charged bodies, due tointeractions among the bodies, charges may get redistributed but it isfound that the total charge of the isolated system is always conserved .

Conservation of charge has been established experimentally.It is not possible to create or destroy net charge carried by any isolated

system although the charge carrying particles may be created or destroyedin a process. Sometimes nature creates charged particles: a neutron turnsinto a proton and an electron. The proton and electron thus created haveequal and opposite charges and the total charge is zero before and after the creation.

1.5.3 Quantisation of charge

Experimentally it is established that all free charges are integral multiplesof a basic unit of charge denoted by e . Thus charge q on a body is always

given by q = ne

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where n is any integer, positive or negative. This basic unit of charge is

the charge that an electron or proton carries. By convention, the chargeon an electron is taken to be negative; therefore charge on an electron is

written as –e and that on a proton as +e. The fact that electric charge is always an integral multiple ofe is termed

as quantisation of charge . There are a large number of situations in physics where certain physical quantities are quantised. The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday. It was experimentally demonstrated by Millikan in 1912.

In the International System (SI) of Units, a unit of charge is called a coulomb and is denoted by the symbol C. A coulomb is defined in terms

the unit of the electric current which you are going to learn in a subsequent chapter. In terms of this definition, one coulomb is the chargeflowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2of Class XI, Physics Textbook , Part I). In this system, the value of the

basic unit of charge is

e = 1.602192 × 10 –19 C

Thus, there are about 6 × 1018 electrons in a charge of –1C. Inelectrostatics, charges of this large magnitude are seldom encounteredand hence we use smaller units 1 µC (micro coulomb) = 10 –6 C or 1 mC(milli coulomb) = 10 –3 C.

If the protons and electrons are the only basic charges in the universe,all the observable charges have to be integral multiples of e . Thus, if a

body contains n 1 electrons and n 2 protons, the total amount of chargeon the body is n 2 × e + n 1 × (–e) = (n 2 – n 1 ) e. Since n 1 and n 2 are integers,their difference is also an integer. Thus the charge on any body is alwaysan integral multiple of e and can be increased or decreased also in stepsof e .

The step size e is, however, very small because at the macroscopiclevel, we deal with charges of a few µC. At this scale the fact that charge of a body can increase or decrease in units of e is not visible. The grainy nature of the charge is lost and it appears to be continuous.

This situation can be compared with the geometrical concepts of points

and lines. A dotted line viewed from a distance appears continuous tous but is not continuous in reality. As many points very close toeach other normally give an impression of a continuous line, many small charges taken together appear as a continuous chargedistribution.

At the macroscopic level, one deals with charges that are enormouscompared to the magnitude of charge e . Since e = 1.6 × 10 –19 C, a chargeof magnitude, say 1 µC, contains something like 1013 times the electroniccharge. At this scale, the fact that charge can increase or decrease only inunits of e is not very different from saying that charge can take continuous

values. Thus, at the macroscopic level, the quantisation of charge has no

practical consequence and can be ignored. At the microscopic level, wherethe charges involved are of the order of a few tens or hundreds of e , i.e.,

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they can be counted, they appear in discrete lumps and quantisation of

charge cannot be ignored. It is the scale involved that is very important.

Example 1.2 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 Con the other body?

Solution In one second 109 electrons move out of the body. Thereforethe charge given out in one second is 1.6 × 10 –19 × 109 C = 1.6 × 10 –10 C. The time required to accumulate a charge of 1 C can then be estimatedto be 1 C ÷ (1.6 × 10 –10 C/s ) = 6.25 × 109 s = 6.25 × 109 ÷ (365 × 24 ×3600) years = 198 years. Thus to collect a charge of one coulomb,from a body from which 109 electrons move out every second, we willneed approximately 200 years. One coulomb is, therefore, a very large

unit for many practical purposes.It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material. A cubic piece of copper of side 1 cm contains about 2.5 × 1024

electrons.

Example 1.3 How much positive and negative charge is there in a cup of water?

Solution Let us assume that the mass of one cup of water is250 g. The molecular mass of water is 18g. Thus, one mole(= 6.02 × 1023 molecules) of water is 18 g. Therefore the number of molecules in one cup of water is (250/18) × 6.02 × 1023.

Each molecule of water contains two hydrogen atoms and one oxygenatom, i.e., 10 electrons and 10 protons. Hence the total positive andtotal negative charge has the same magnitude. It is equal to(250/18) × 6.02 × 1023 × 10 × 1.6 × 10 –19 C = 1.34 × 107 C.

1.6 COULOMB’S L AW Coulomb’s law is a quantitative statement about the force between twopoint charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and thecharged bodies are treated as point charges . Coulomb measured theforce between two point charges and found that it varied inversely as the square of the distance between the charges and was directly

proportional to the product of the magnitude of the two charges and

acted along the line joining the two charges . Thus, if two point chargesq 1, q 2 are separated by a distance r in vacuum, the magnitude of theforce (F ) between them is given by

212

q q F k

r = (1.1)

How did Coulomb arrive at this law from his experiments? Coulombused a torsion balance* for measuring the force between two charged metallic

* A torsion balance is a sensitive device to measure force. It was also used later by Cavendish to measure the very feeble gravitational force between two objects,to verify Newton’s Law of Gravitation.

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spheres. When the separation between two spheres is much

larger than the radius of each sphere, the charged spheresmay be regarded as point charges. However, the chargeson the spheres were unknown, to begin with. How thencould he discover a relation like Eq. (1.1)? Coulombthought of the following simple way: Suppose the chargeon a metallic sphere is q . If the sphere is put in contact

with an identical uncharged sphere, the charge will spreadover the two spheres. By symmetry, the charge on eachsphere will be q /2*. Repeating this process, we can get charges q /2, q /4, etc. Coulomb varied the distance for a fixed pair of charges and measured the force for different

separations. He then varied the charges in pairs, keepingthe distance fixed for each pair. Comparing forces for different pairs of charges at different distances, Coulombarrived at the relation, Eq. (1.1).

Coulomb’s law, a simple mathematical statement, was initially experimentally arrived at in the manner described above. While the original experimentsestablished it at a macroscopic scale, it has also beenestablished down to subatomic level (r ~ 10 –10 m).

Coulomb discovered his law without knowing theexplicit magnitude of the charge. In fact, it is the other

way round: Coulomb’s law can now be employed to

furnish a definition for a unit of charge. In the relation,Eq. (1.1), k is so far arbitrary. We can choose any positive

value of k . The choice of k determines the size of the unit of charge. In SI units, the value of k is about 9 × 109.

The unit of charge that results from this choice is calleda coulomb which we defined earlier in Section 1.4.Putting this value of k in Eq. (1.1), we see that for q 1 = q 2 = 1 C, r = 1 m

F = 9 × 109 N That is, 1 C is the charge that when placed at a

distance of 1 m from another charge of the same

magnitude in vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N. One coulomb isevidently too big a unit to be used. In practice, inelectrostatics, one uses smaller units like 1 mC or 1 µC.

The constant k in Eq. (1.1) is usually put ask = 1/4πε 0 for later convenience, so that Coulomb’s law is written as

0

1 22

14

q q F

r ε =

π(1.2)

ε 0 is called the permittivity of free space . The value of ε 0 in SI units is

0ε = 8.854 × 10 –12 C2 N –1m –2

* Implicit in this is the assumption of additivity of charges and conservation:two charges (q /2 each) add up to make a total charge q.

Charles Augustin deCoulomb (1736 – 1806)

Coulomb, a Frenchphysicist, began his career as a military engineer inthe West Indies. In 1776, hereturned to Paris andretired to a small estate todo his scientific research.He invented a torsion balance to measure thequantity of a force and used

it for determination of forces of electric attractionor repulsion between smallcharged spheres. He thusarrived in 1785 at theinverse square law relation,now known as Coulomb’slaw. The law had beenanticipated by Priestley andalso by Cavendish earlier,though Cavendish never published his results.Coulomb also found the

inverse square law of force be tween un like and likemagnetic poles.

CHARL E S A U G U S T I ND

E C O UL OMB

( 1 7 3 6 –1

8 0 6 )

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Since force is a vector, it is better to write

Coulomb’s law in the vector notation. Let theposition vectors of charges q 1 and q 2 be r1 and r2respectively [see Fig.1.6(a)]. We denote force onq 1 due to q 2 by F 12 and force on q 2 due to q 1 by F 21. The two point charges q 1 and q 2 have beennumbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21:

r21 = r2 – r1

In the same way, the vector leading from 2 to1 is denoted by r12:

r12 = r1 – r2 = – r21

The magnitude of the vectors r21 and r12 isdenoted by r 21 and r 12, respectively (r 12 = r 21). Thedirection of a vector is specified by a unit vector along the vector. To denote the direction from 1to 2 (or from 2 to 1), we define the unit vectors:

2121

21

ˆr

=r

r ,12

12 21 1212

ˆ ˆ ˆ,r

= =r

r r r

Coulomb’s force law between two point charges q 1 and q 2 located at r1 and r2 is then expressed as

1 221 212

21

1ˆ

4 o

q q

r ε =

πF r (1.3)

Some remarks on Eq. (1.3) are relevant:

• Equation (1.3) is valid for any sign of q 1 and q 2 whether positive or negative. If q 1 and q 2 are of the same sign (either both positive or bothnegative), F 21 is along r̂ 21, which denotes repulsion, as it should be for like charges. If q 1 and q 2 are of opposite signs, F 21 is along – ̂r 21(= r̂ 12),

which denotes attraction, as expected for unlike charges. Thus, we donot have to write separate equations for the cases of like and unlikecharges. Equation (1.3) takes care of both cases correctly [Fig. 1.6(b)].

• The forceF 12 on charge q 1 due to charge q 2, is obtained from Eq. (1.3), by simply interchanging 1 and 2, i.e.,

1 212 12 212

0 12

1ˆ

4q q

r ε = = −

πF r F

Thus, Coulomb’s law agrees with the Newton’s third law.

• Coulomb’s law [Eq. (1.3)] gives the force between two charges q 1 andq 2 in vacuum. If the charges are placed in matter or the interveningspace has matter, the situation gets complicated due to the presence

of charged constituents of matter. We shall consider electrostatics inmatter in the next chapter.

FIGURE 1.6 (a) Geometry and(b) Forces between charges.

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Example 1.4 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between twostationary point masses, both have inverse-square dependence onthe distance between the charges/masses. (a) Compare the strengthof these forces by determining the ratio of their magnitudes (i) for anelectron and a proton and (ii) for two protons. (b) Estimate theaccelerations of electron and proton due to the electrical force of their mutual attraction when they are 1 Å (= 10-10 m) apart? (m

p = 1.67 ×

10 –27 kg, m e = 9.11 × 10 –31 kg)

Solution(a) (i) The electric force between an electron and a proton at a distance

r apart is:2

20

1

4e e

F r ε = −

π where the negative sign indicates that the force is attractive. Thecorresponding gravitational force (always attractive) is:

2 p e

G

m m F G

r = −

where m p and m

e are the masses of a proton and an electron

respectively.2

39

0

2.4 104

e

G p e

F e

F Gm m ε = = ×

π

(ii) On similar lines, the ratio of the magnitudes of electric forceto the gravitational force between two protons at a distance r

apart is :2

04e

G p p

F e

F Gm m ε = =

π 1.3 × 1036

However, it may be mentioned here that the signs of the two forcesare different. For two protons, the gravitational force is attractivein nature and the Coulomb force is repulsive . The actual valuesof these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) are F e ~ 230 N whereas F G ~ 1.9 × 10

–34 N. The (dimensionless) ratio of the two forces shows that electricalforces are enormously stronger than the gravitational forces.

(b) The electric force F exerted by a proton on an electron is same inmagnitude to the force exerted by an electron on a proton; however

the masses of an electron and a proton are different. Thus, themagnitude of force is

|F | = 2

20

14

e

r ε π = 8.987 × 109 Nm2/C2 × (1.6 ×10 –19C)2 / (10 –10m)2

= 2.3 × 10 –8 NUsing Newton’s second law of motion, F = ma , the accelerationthat an electron will undergo isa = 2.3×10 –8 N / 9.11 ×10 –31 kg = 2.5 × 1022 m/s2

Comparing this with the value of acceleration due to gravity, wecan conclude that the effect of gravitational field is negligible onthe motion of electron and it undergoes very large accelerationsunder the action of Coulomb force due to a proton.

The value for acceleration of the proton is2.3 × 10 –8 N / 1.67 × 10 –27 kg = 1.4 × 1019 m/s2

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Example 1.5 A charged metallic sphere A is suspended by a nylonthread. Another charged metallic sphere B held by an insulatinghandle is brought close to A such that the distance between their centres is 10 cm, as shown in Fig. 1.7(a). The resulting repulsion of A is noted (for example, by shining a beam of light and measuring thedeflection of its shadow on a screen). Spheres A and B are touched by uncharged spheres C and D respectively, as shown in Fig. 1.7(b).C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in Fig. 1.7(c). What is the expected repulsion of A on the basis of Coulomb’s law?Spheres A and C and spheres B and D have identical sizes. Ignorethe sizes of A and B in comparison to the separation between their centres.

FIGURE 1.7

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Solution Let the original charge on sphere A be q and that on B beq ′. At a distance r between their centres, the magnitude of theelectrostatic force on each is given by

20

14

qq F

r ε

′=

π

neglecting the sizes of spheres A and B in comparison to r . When anidentical but uncharged sphere C touches A, the charges redistributeon A and C and, by symmetry, each sphere carries a charge q /2.Similarly, after D touches B, the redistributed charge on each isq ′/2. Now, if the separation between A and B is halved, the magnitudeof the electrostatic force on each is

2 20 0

1 ( /2)( /2) 1 ( )4 4( /2)

q q qq F F

r r ε ε ′ ′= = =′

π π

Thus the electrostatic force on A, due to B, remains unaltered.

1.7 F ORCES BETWEEN MULTIPLE CHARGES The mutual electric force between two charges is given by Coulomb’s law. How to calculate the force on a charge where there are not one but several chargesaround? Consider a system of n stationary charges

q 1, q 2, q 3, ..., q n in vacuum. What is the force on q 1 dueto q 2, q 3, ..., q n ? Coulomb’s law is not enough to answer this question. Recall that forces of mechanical originadd according to the parallelogram law of addition. Isthe same true for forces of electrostatic origin?

Experimentally it is verified that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces

are unaffected due to the presence of other charges . This is termed as the principle of superposition .

To better understand the concept, consider a

system of three charges q 1, q 2 and q 3, as shown inFig. 1.8(a). The force on one charge, say q 1, due to twoother charges q 2, q 3 can therefore be obtained by performing a vector addition of the forces due to eachone of these charges. Thus, if the force on q 1 due to q 2is denoted by F 12, F 12 is given by Eq. (1.3) even thoughother charges are present.

Thus, F 121 2

1220 12

1ˆ

4q q

r ε =

πr

In the same way, the force on q 1 due to q 3, denoted by F 13, is given by

1 313 132

0 13

1 ˆ4

q q r ε

=π

F r

FIGURE 1.8 A system of (a) threecharges (b) multiple charges.

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which again is the Coulomb force on q 1 due to q 3, even though other

charge q 2 is present. Thus the total force F 1 on q 1 due to the two charges q 2 and q 3 is

given as

1 31 21 12 13 12 132 2

0 012 13

1 1ˆ ˆ

4 4

q q q q

r r ε ε = + = +

π πF F F r r (1.4)

The above calculation of force can be generalised to a system of charges more than three, as shown in Fig. 1.8(b).

The principle of superposition says that in a system of charges q 1,q 2, ..., q n , the force on q 1 due to q 2 is the same as given by Coulomb’s law,i.e., it is unaffected by the presence of the other charges q 3, q 4, ..., q n . The

total force F 1 on the charge q 1, due to all other charges, is then given by the vector sum of the forces F 12, F 13, ..., F 1n :

i.e.,

1 3 11 21 12 13 1n 12 13 12 2 2

0 12 13 1

1ˆ ˆ ˆ= + + ...+ ...

4n

n

n

q q q q q q

r r r ε

⎡ ⎤= + + +⎢ ⎥

π ⎣ ⎦F F F F r r r

112

20 1

ˆ4

n i

i

i i

q q

r ε ==

π ∑ r (1.5)

The vector sum is obtained as usual by the parallelogram law of addition of vectors. All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.

Example 1.6 Consider three charges q 1, q 2, q 3 each equal to q at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q ) placed at the centroid of thetriangle, as shown in Fig. 1.9?

FIGURE 1.9

Solution In the given equilateral triangle ABC of sides of length l , if we draw a perpendicular AD to the side BC,

AD = AC cos 30º = ( 3 /2 ) l and the distance AO of the centroid Ofrom A is (2/3) AD = (1/ 3 ) l. By symmatry AO = BO = CO.

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EXAMP L E1 . 6

Thus,

Force F 1 on Q due to charge q at A = 20

34

Qq

l ε π along AO

Force F 2 on Q due to charge q at B = 20

34

Qq

l ε π along BO

Force F 3 on Q due to charge q at C = 20

34

Qq

l ε π along CO

The resultant of forces F 2 and F 3 is 20

34

Qq

l ε π along OA, by the

parallelogram law. Therefore, the total force on Q = ( )20

3ˆ ˆ

4

Qq

l ε

−

π

r r

= 0, where r̂ is the unit vector along OA.It is clear also by symmetry that the three forces will sum to zero.Suppose that the resultant force was non-zero but in some direction.Consider what would happen if the system was rotated through 60ºabout O.

Example 1.7 Consider the charges q , q, and – q placed at the verticesof an equilateral triangle, as shown in Fig. 1.10. What is the force oneach charge?

FIGURE 1.10

Solution The forces acting on charge q at A due to charges q at Band – q at C are F 12 along BA and F 13 along AC respectively, as shownin Fig. 1.10. By the parallelogram law, the total force F 1 on the chargeq at A is given by

F 1 = F 1̂r where 1̂r is a unit vector along BC. The force of attraction or repulsion for each pair of charges has the

same magnitude2

204

q F

l ε =

π

The total force F 2 on charge q at B is thus F 2 = F r̂ 2, where r̂ 2 is a unit vector along AC.

EXAMP L E

1 .7

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Similarly the total force on charge – q at C is F 3 = 3 F ˆn , where n̂ isthe unit vector along the direction bisecting the ∠BCA.

It is interesting to see that the sum of the forces on the three chargesis zero, i.e.,

F 1 + F 2 + F 3 = 0

The result is not at all surprising. It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law. The proof is left to you as an exercise.

1.8 ELECTRIC F IELDLet us consider a point charge Q placed in vacuum, at the origin O. If we

place another point charge q at a point P, where OP = r, then the charge Q will exert a force on q as per Coulomb’s law. We may ask the question: If charge q is removed, then what is left in the surrounding? Is therenothing? If there is nothing at the point P, then how does a force act

when we place the charge q at P. In order to answer such questions, theearly scientists introduced the concept of field . According to this, we say that the charge Q produces an electric field everywhere in the surrounding.

When another charge q is brought at some point P, the field there acts onit and produces a force. The electric field produced by the charge Q at a point r is given as

( )2 20 0

1 1ˆ ˆ

4 4

Q Q

r r ε ε = =

π π

E r r r

(1.6) where ˆ =r r/r, is a unit vector from the origin to the point r. Thus, Eq.(1.6)specifies the value of the electric field for each value of the position

vector r. The word “field” signifies how some distributed quantity (whichcould be a scalar or a vector) varies with position. The effect of the chargehas been incorporated in the existence of the electric field. We obtain theforce F exerted by a charge Q on a charge q , as

20

1ˆ

4Qq

r ε =

πF r (1.7)

Note that the charge q also exerts an equal and opposite force on thecharge Q. The electrostatic force between the charges Q and q can be

looked upon as an interaction between charge q and the electric field of Q and vice versa. If we denote the position of charge q by the vector r, it experiences a force F equal to the charge q multiplied by the electricfield E at the location of q. Thus,

F (r) = q E(r) (1.8)Equation (1.8) defines the SI unit of electric field as N/C*.Some important remarks may be made here:

(i) From Eq. (1.8), we can infer that if q is unity, the electric field due toa charge Q is numerically equal to the force exerted by it. Thus, theelectric field due to a charge Q at a point in space may be defined as the force that a unit positive charge would experience if placed

* An alternate unit V/m will be introduced in the next chapter.

FIGURE 1.11 Electricfield (a) due to a

charge Q , (b) due to a

charge–Q

.

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at that point . The charge Q, which is producing the electric field, is

called a source charge and the charge q, which tests the effect of a source charge, is called a test charge . Note that the source charge Q must remain at its original location. However, if a charge q is brought at any point around Q , Q itself is bound to experience an electricalforce due to q and will tend to move. A way out of this difficulty is tomake q negligibly small. The force F is then negligibly small but theratio F /q is finite and defines the electric field:

0limq q →

⎛ ⎞= ⎜ ⎟⎝ ⎠

F E (1.9)

A practical way to get around the problem (of keepingQ undisturbed

in the presence of q ) is to hold Q to its location by unspecified forces! This may look strange but actually this is what happens in practice. When we are considering the electric force on a test charge q due to a charged planar sheet (Section 1.15), the charges on the sheet are held totheir locations by the forces due to the unspecified charged constituentsinside the sheet.(ii) Note that the electric field E due to Q , though defined operationally

in terms of some test charge q, is independent of q . This is becauseF is proportional to q , so the ratio F /q does not depend on q . Theforce F on the charge q due to the charge Q depends on the particular location of charge q which may take any value in the space around

the charge Q. Thus, the electric field E due to Q is also dependent onthe space coordinate r. For different positions of the charge q all over the space, we get different values of electric field E. The field exists at every point in three-dimensional space.

(iii) For a positive charge, the electric field will be directed radially outwards from the charge. On the other hand, if the source charge isnegative, the electric field vector, at each point, points radially inwards.

(iv) Since the magnitude of the force F on charge q due to charge Q depends only on the distance r of the charge q from charge Q ,the magnitude of the electric field E will also depend only on thedistance r . Thus at equal distances from the charge Q , the magnitude

of its electric field E is same. The magnitude of electric field E due toa point charge is thus same on a sphere with the point charge at itscentre; in other words, it has a spherical symmetry.

1.8.1 Electric field due to a system of charges

Consider a system of charges q 1, q 2, ..., q n with position vectors r1,r2, ..., rn relative to some origin O. Like the electric field at a point inspace due to a single charge, electric field at a point in space due to thesystem of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the originalpositions of charges q 1, q 2, ..., q n . We can use Coulomb’s law and the

superposition principle to determine this field at a point P denoted by position vector r.

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Electric field E1 at r due to q 1 at r1 is given by

E1 =1

1P2

0 1P

1ˆ

4q

r πεr

where 1Pr̂ is a unit vector in the direction from q 1 to P,and r 1P is the distance between q 1 and P.In the same manner, electric field E2 at r due to q 2 at r2 is

E2 =2

2P2

0 2P

1ˆ

4q

r πεr

where 2Pr̂ is a unit vector in the direction from q 2 to P

and r 2P is the distance between q 2 and P. Similar expressions hold good for fields E3, E4, ..., En due tocharges q 3, q 4, ..., q n .By the superposition principle, the electric field E at rdue to the system of charges is (as shown in Fig. 1.12)

E(r) = E1 (r) + E2 (r) + … + En(r)

= 1 21P 2P P2 2 20 0 01P 2P P

1 1 1ˆ ˆ ˆ...

4 4 4n

n

n

q q q

r r r ε ε ε + + +

π π πr r r

E(r) i P210 P

1ˆ

4

n i

i i

q

r ε ==

π ∑ r (1.10)

E is a vector quantity that varies from one point

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Physics 1 - Class XII